Formules de Feynmann-Kac pour le modèle de Nelson ultra-violet renormalisée

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
O. Matte, J. S. Møller
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引用次数: 20

Abstract

We derive Feynman-Kac formulas for the ultra-violet renormalized Nelson Hamiltonian with a Kato decomposable external potential and for corresponding fiber Hamiltonians in the translation invariant case. We simultaneously treat massive and massless bosons. Furthermore, we present a non-perturbative construction of a renormalized Nelson Hamiltonian in the non-Fock representation defined as the generator of a corresponding Feynman-Kac semi-group. Our novel analysis of the vacuum expectation of the Feynman-Kac integrands shows that, if the external potential and the Pauli-principle are dropped, then the spectrum of the $N$-particle renormalized Nelson Hamiltonian is bounded from below by some negative universal constant times $g^4N^3$, for all values of the coupling constant $g$. A variational argument also yields an upper bound of the same form for large $g^2N$. We further verify that the semi-groups generated by the ultra-violet renormalized Nelson Hamiltonian and its non-Fock version are positivity improving with respect to a natural self-dual cone, if the Pauli principle is ignored. In another application we discuss continuity properties of elements in the range of the semi-group of the renormalized Nelson Hamiltonian.
重整化Nelson紫外模型的Feynmann-Kac公式
我们导出了具有加托可分解外势的紫外重整化纳尔逊哈密顿量的费曼-卡茨公式以及平移不变情况下相应的纤维哈密顿量的费曼-卡茨公式。我们同时处理有质量和无质量玻色子。进一步,我们给出了非fock表示中的重规格化Nelson hamilton算子的非摄动构造,该构造被定义为相应的Feynman-Kac半群的生成器。我们对Feynman-Kac积分的真空期望的新分析表明,如果放弃外势和保利原理,那么对于所有耦合常数g$, N$粒子重归一化的纳尔逊哈密顿量的谱从下面被一个负的通用常数乘以g^4N^3$所限制。对于较大的$g^2N$,变分参数也给出了相同形式的上界。在忽略泡利原理的情况下,我们进一步验证了由紫外重正化Nelson hamilton算子及其非fock算子生成的半群对于自然自对偶锥是正改进的。在另一个应用中,我们讨论了重归一化纳尔逊哈密顿算子半群范围内元素的连续性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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